G Tewani Solutions for Chapter: Different Products of Vectors and Their Geometrical Applications, Exercise 1: DPP

The vectors $\overrightarrow{x}$ and $\overrightarrow{y}$ satisfy the equation $p\overrightarrow{x}+q\overrightarrow{y}=\overrightarrow{a}$, (where $p, q$ are scalars and $\overrightarrow{a}$ is a known vector).

It is given that $\overrightarrow{x}·\overrightarrow{y}\ge \frac{{\left|\overrightarrow{a}\right|}^2}{4pq}$, then $\frac{\left|\overrightarrow{x}\right|}{\left|\overrightarrow{y}\right|}$ is equal to, $\left(pq>0\right)$

If $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ non-zero vectors such that $\overrightarrow{a}$ is perpendicular to $\overrightarrow{b}$ and $\overrightarrow{c}$ and $\left|\overrightarrow{a}\right|=1, \left|\overrightarrow{b}\right|=2, \left|\overrightarrow{c}\right|=1, \overrightarrow{b}·\overrightarrow{c}=1$. There is a non-zero vector $\overrightarrow{d}$ coplanar with $\overrightarrow{a}+\overrightarrow{b}$ and  $2\overrightarrow{b}-\overrightarrow{c}$ and $\overrightarrow{d}·\overrightarrow{a}=1$, then the minimum value of $\left|\overrightarrow{d}\right|$ is

Let two non-collinear vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ inclined at an angle $\frac{2\pi }{3}$ be such that  $\left|\overrightarrow{a}\right|=3$ and $\left|\overrightarrow{b}\right|=2$. If a point $P$ moves so that at any time $t$ its position vector $\overrightarrow{OP}$, (where $O$ is the origin) is given as $\overrightarrow{OP}=\left(t+\frac{1}{t}\right)\overrightarrow{a}+\left(t-\frac{1}{t}\right)\overrightarrow{b}$, then least distance of $P$ from the origin is

Four vectors $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ and $\overrightarrow{\mathrm{x}}$ satisfy the relation $\left(\overrightarrow{\mathrm{a}}·\overrightarrow{\mathrm{x}}\right)\overrightarrow{\mathrm{b}}=\overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{x}}$, where $\overrightarrow{\mathrm{a}}·\overrightarrow{\mathrm{b}}\ne 1$. The value of $\overrightarrow{\mathrm{x}}$ in terms of $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}$ and $\overrightarrow{c}$ is equal to

If the area of a triangular face $BCD$, of a regular tetrahedron $ABCD$ is $4\sqrt{3}$ sq.units, then the area of a triangle whose two sides are represented by the vectors $\overrightarrow{AB}$ and $\overrightarrow{CD}$ is

If $OABC$ is a tetrahedron such that $O{A}^2+B{C}^2=O{B}^2+C{A}^2=O{C}^2+A{B}^2,$ then which of the following is not true?

If $\overrightarrow{a}$, $\overrightarrow{b}$, and $\overrightarrow{c}$ are three unit vectors equally inclined to each other at an angle $\alpha$, then the angle between $\overrightarrow{a}$ and the plane of $\overrightarrow{b}$ and $\overrightarrow{c}$ is

If $a$, $b$, $c$ and $A$, $B$, $C\in R-\left\{0\right\}$ such that, $aA+bB+cC+\sqrt{\left(a^2+b^2+c^2\right)\left(A^2+B^2+C^2\right)}=0$, then the value of $\frac{aB}{bA}+\frac{bC}{cB}+\frac{cA}{aC}$ is